Wilcoxon Test
The t-tests reviewed in the previous chapter are
suitable for studies with normally distributed results. However, if
there are outliers, then the t-tests are not sensitive and
non-parametric tests have to be applied. We should add that
non-parametric are also adequate for testing normally distributed
data. And, so, these tests are, actually, universal, and are,
therefore, absolutely to be recommended.
Calculate the p-value with the paired Wilcoxon
test.
|
Observation 1:
|
|||||||||
|
6.0,
|
7.1,
|
8.1,
|
7.5,
|
6.4,
|
7.9,
|
6.8,
|
6.6,
|
7.3,
|
5.6
|
|
Observation 2:
|
|||||||||
|
5.1,
|
8.0,
|
3.8,
|
4.4,
|
5.2,
|
5.4,
|
4.3,
|
6.0,
|
7.3,
|
6.2
|
|
Individual differences:
|
|||||||||
|
0.9,
|
−0.9,
|
4.3,
|
3.1,
|
1.2,
|
2.5,
|
2.5,
|
0.6,
|
3.6,
|
−0.6
|
|
Rank number:
|
|||||||||
|
3.5,
|
3.5,
|
10,
|
7,
|
5,
|
8,
|
6,
|
2,
|
9,
|
1
|
A.
not significant
B.
0.05 < p < 0.10
C.
p < 0.05
D.
P < 0.01
Is there a significant difference between
observation 1 and 2? Which significance level is correct?
The individual differences are given a rank
number dependent on their magnitude of difference. If two
differences are identical, and if they have for example the rank
numbers 3 and 4, then an average rank number is given to both of
them, which means 3.5 and 3.5. Next, all positive and all negative
rank numbers have to be added up separately. We will find 4.5 and
50.5. According to the Wilcoxon table underneath the smaller one of
the two add-up numbers must be smaller than 8 in order to be able
to speak of a p-value <0.05. This is true in our example.
Wilcoxon test table
|
Number of pairs
|
P < 0.05
|
P < 0.01
|
|---|---|---|
|
7
|
2
|
0
|
|
8
|
2
|
0
|
|
9
|
6
|
2
|
|
10
|
8
|
3
|
|
11
|
11
|
5
|
|
12
|
14
|
7
|
|
13
|
17
|
10
|
|
14
|
21
|
13
|
|
15
|
25
|
16
|
|
16
|
30
|
19
|
Mann-Whitney Test
Like the Wilcoxon test, being the non-parametric
alternative for the paired t-test, the Mann-Whitney test is the
non-parametric alternative for the unpaired t-test. Also this test
is applicable for all kinds of data, and, therefore, particularly,
to be recommended for investigators with little affection for
medical statistics.
Calculate the p-value of the difference between
two groups of ten patients with the help of this test.
|
Group 1:
|
|||||||||
|
6.0
|
7.1,
|
8.1,
|
7.5,
|
6.4,
|
7.9,
|
6.8,
|
6.6,
|
7.3,
|
5.6
|
|
Group 2:
|
|||||||||
|
5.1,
|
8.0,
|
3.8,
|
4.4,
|
5.2,
|
5.4,
|
4.3,
|
6.0,
|
3.7,
|
6.2
|
A.
not significant
B.
0.05 < p < 0.10
C.
p < 0.05
D.
p < 0.01
Is there a significant difference between the two
groups? What significance level is correct?
All values are ranked together in ascending order
of magnitude. The values from group 1 are printed thin, those from
group 2 are printed fat. Add a rank number to each value. If there
are identical values, for example, the rank numbers 9 and 10, then
replace those rank numbers with average rank numbers, 9.5 and
9.5.
Subsequently, all fat printed rank numbers are
added up, and so are the thin printed rank numbers. We will find
the values 142.5 for fat print, and 67.5 for thin print.
According to the Mann-Whitney table of page 13,
the difference should be larger than 71 in order for the
significance level of difference to be <0.05. We find a
difference of 75, which means that there is a p-value <0.05 and
that the difference between the two groups is, thus, significant.
|
3.7
|
1
|
|
3.8
|
2
|
|
4.3
|
3
|
|
4.4
|
4
|
|
5.1
|
5
|
|
5.2
|
6
|
|
5.4
|
7
|
|
5.6
|
8
|
|
6.0
|
9.5
|
|
6.0
|
9.5
|
|
6.2
|
11
|
|
6.4
|
12
|
|
6.6
|
13
|
|
6.8
|
14
|
|
7.1
|
15
|
|
7.3
|
16
|
|
7.5
|
17
|
|
7.9
|
18
|
|
8.0
|
19
|
|
8.1
|
20
|
Mann-Whitney test
|
P < 0.01 levels
|
|||||||||||||||
|
n1→
|
|||||||||||||||
|
n2
↓
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
10
|
11
|
12
|
13
|
14
|
15
|
|
|
4
|
10
|
||||||||||||||
|
5
|
6
|
11
|
17
|
||||||||||||
|
6
|
7
|
12
|
18
|
26
|
|||||||||||
|
7
|
7
|
13
|
20
|
27
|
36
|
||||||||||
|
8
|
3
|
8
|
14
|
21
|
29
|
38
|
49
|
||||||||
|
9
|
3
|
8
|
15
|
22
|
31
|
40
|
51
|
63
|
|||||||
|
10
|
3
|
9
|
15
|
23
|
32
|
42
|
53
|
65
|
78
|
||||||
|
11
|
4
|
9
|
16
|
24
|
34
|
44
|
55
|
68
|
81
|
96
|
|||||
|
12
|
4
|
10
|
17
|
26
|
35
|
46
|
58
|
71
|
85
|
99
|
115
|
||||
|
13
|
4
|
10
|
18
|
27
|
37
|
48
|
60
|
73
|
88
|
103
|
119
|
137
|
|||
|
14
|
4
|
11
|
19
|
28
|
38
|
50
|
63
|
76
|
91
|
106
|
123
|
141
|
160
|
||
|
15
|
4
|
11
|
20
|
29
|
40
|
52
|
65
|
79
|
94
|
110
|
127
|
145
|
164
|
185
|
|
|
16
|
4
|
12
|
21
|
31
|
42
|
54
|
67
|
82
|
97
|
114
|
131
|
150
|
169
|
||
|
17
|
5
|
12
|
21
|
32
|
43
|
56
|
70
|
84
|
100
|
117
|
135
|
154
|
|||
|
18
|
5
|
13
|
22
|
33
|
45
|
58
|
72
|
87
|
103
|
121
|
139
|
||||
|
19
|
5
|
13
|
23
|
34
|
46
|
60
|
74
|
90
|
107
|
124
|
|||||
|
20
|
5
|
14
|
24
|
35
|
48
|
62
|
77
|
93
|
110
|
||||||
|
21
|
6
|
14
|
25
|
37
|
50
|
64
|
79
|
95
|
|||||||
|
22
|
6
|
15
|
26
|
38
|
51
|
66
|
82
|
||||||||
|
23
|
6
|
15
|
27
|
39
|
53
|
68
|
|||||||||
|
24
|
6
|
16
|
28
|
40
|
55
|
||||||||||
|
25
|
6
|
16
|
28
|
42
|
|||||||||||
|
26
|
7
|
17
|
29
|
||||||||||||
|
27
|
7
|
17
|
|||||||||||||
|
28
|
7
|
||||||||||||||
Mann-Whitney test
|
P < 0.05 levels
|
|||||||||||||||
|
n1→
|
|||||||||||||||
|
n2
↓
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
10
|
11
|
12
|
13
|
14
|
15
|
|
|
5
|
15
|
||||||||||||||
|
6
|
10
|
16
|
23
|
||||||||||||
|
7
|
10
|
17
|
24
|
32
|
|||||||||||
|
8
|
11
|
17
|
25
|
34
|
43
|
||||||||||
|
9
|
6
|
11
|
18
|
26
|
35
|
45
|
56
|
||||||||
|
10
|
6
|
12
|
19
|
27
|
37
|
47
|
58
|
71
|
|||||||
|
11
|
6
|
12
|
20
|
28
|
38
|
49
|
61
|
74
|
87
|
||||||
|
12
|
7
|
13
|
21
|
30
|
40
|
51
|
63
|
76
|
90
|
106
|
|||||
|
13
|
7
|
14
|
22
|
31
|
41
|
53
|
65
|
79
|
93
|
109
|
125
|
||||
|
14
|
7
|
14
|
22
|
32
|
43
|
54
|
67
|
81
|
96
|
112
|
129
|
147
|
|||
|
15
|
8
|
15
|
23
|
33
|
44
|
56
|
70
|
84
|
99
|
115
|
133
|
151
|
171
|
||
|
16
|
8
|
15
|
24
|
34
|
46
|
58
|
72
|
86
|
102
|
119
|
137
|
155
|
|||
|
17
|
8
|
16
|
25
|
36
|
47
|
60
|
74
|
89
|
105
|
122
|
140
|
||||
|
18
|
8
|
16
|
26
|
37
|
49
|
62
|
76
|
92
|
108
|
125
|
|||||
|
19
|
3
|
9
|
17
|
27
|
38
|
50
|
64
|
78
|
94
|
111
|
|||||
|
20
|
3
|
9
|
18
|
28
|
39
|
52
|
66
|
81
|
97
|
||||||
|
21
|
3
|
9
|
18
|
29
|
40
|
53
|
68
|
83
|
|||||||
|
22
|
3
|
10
|
19
|
29
|
42
|
55
|
70
|
||||||||
|
23
|
3
|
10
|
19
|
30
|
43
|
57
|
|||||||||
|
24
|
3
|
10
|
20
|
31
|
44
|
||||||||||
|
25
|
3
|
11
|
20
|
32
|
|||||||||||
|
26
|
3
|
11
|
21
|
||||||||||||
|
27
|
4
|
11
|
|||||||||||||
|
28
|
4
|
||||||||||||||